2022 NYJC P2 Q9
A bag contains three yellow marbles, one blue marble and $x$ red marbles, where $x>1$. In a game, Lily takes $~2$ marbles at random from the bag, without replacement. She scores $~2$ points for each red marble taken and $1$ point for each yellow marble taken. If a blue marble is taken, she loses $~2$ points. The random variable $S$ is the sum of the points of the two marbles taken.
(i)
Find $\text{P}\left( S=s \right)$ for all possible values of $s$.
[4]
(ii)
Show that $\text{E}\left( S \right)=\frac{2\left( 2x+1 \right)}{x+4}$ and find an expression for $\text{Var}\left( S \right)$.
[5]
(iii)
Find the least value of $x$ if $\text{E}\left( S \right)>2$.
[2]
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